Precalculus Examples

${(x - 8)}^{2} + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1

Simplify the left side ${(x - 8)}^{2} + \frac{{(y + 4)}^{2}}{60}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$(x - 8) (x - 8) + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$x (x - 8) - 8 (x - 8) + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 8 - 8 (x - 8) + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.3.1.2

Move $- 8$ to the left of $x$ .

$x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.3.1.3

Multiply $- 8$ by $- 8$ .

$x^{2} - 8 x - 8 x + 64 + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.1.3.2

Subtract $8 x$ from $- 8 x$ .

$x^{2} - 16 x + 64 + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{60}{60}$ .

$- 16 x + 64 + x^{2} \cdot \frac{60}{60} + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $x^{2}$ and $\frac{60}{60}$ .

$- 16 x + 64 + \frac{x^{2} \cdot 60}{60} + \frac{{(y + 4)}^{2}}{60} = 50$

Step 1.3.2

Combine the numerators over the common denominator.

$- 16 x + 64 + \frac{x^{2} \cdot 60 + {(y + 4)}^{2}}{60} = 50$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Move $60$ to the left of $x^{2}$ .

$- 16 x + 64 + \frac{60 \cdot x^{2} + {(y + 4)}^{2}}{60} = 50$

Step 1.4.2

Rewrite ${(y + 4)}^{2}$ as $(y + 4) (y + 4)$ .

$- 16 x + 64 + \frac{60 x^{2} + (y + 4) (y + 4)}{60} = 50$

Step 1.4.3

Expand $(y + 4) (y + 4)$ using the FOIL Method.

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Step 1.4.3.1

Apply the distributive property.

$- 16 x + 64 + \frac{60 x^{2} + y (y + 4) + 4 (y + 4)}{60} = 50$

Step 1.4.3.2

Apply the distributive property.

$- 16 x + 64 + \frac{60 x^{2} + y \cdot y + y \cdot 4 + 4 (y + 4)}{60} = 50$

Step 1.4.3.3

Apply the distributive property.

$- 16 x + 64 + \frac{60 x^{2} + y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4}{60} = 50$

Step 1.4.4

Simplify and combine like terms.

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Step 1.4.4.1

Simplify each term.

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Step 1.4.4.1.1

Multiply $y$ by $y$ .

$- 16 x + 64 + \frac{60 x^{2} + y^{2} + y \cdot 4 + 4 y + 4 \cdot 4}{60} = 50$

Step 1.4.4.1.2

Move $4$ to the left of $y$ .

$- 16 x + 64 + \frac{60 x^{2} + y^{2} + 4 \cdot y + 4 y + 4 \cdot 4}{60} = 50$

Step 1.4.4.1.3

Multiply $4$ by $4$ .

$- 16 x + 64 + \frac{60 x^{2} + y^{2} + 4 y + 4 y + 16}{60} = 50$

Step 1.4.4.2

Add $4 y$ and $4 y$ .

$- 16 x + 64 + \frac{60 x^{2} + y^{2} + 8 y + 16}{60} = 50$

Step 1.5

To write $- 16 x$ as a fraction with a common denominator, multiply by $\frac{60}{60}$ .

$- 16 x \cdot \frac{60}{60} + \frac{60 x^{2} + y^{2} + 8 y + 16}{60} + 64 = 50$

Step 1.6

Simplify terms.

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Step 1.6.1

Combine $- 16 x$ and $\frac{60}{60}$ .

$\frac{- 16 x \cdot 60}{60} + \frac{60 x^{2} + y^{2} + 8 y + 16}{60} + 64 = 50$

Step 1.6.2

Combine the numerators over the common denominator.

$\frac{- 16 x \cdot 60 + 60 x^{2} + y^{2} + 8 y + 16}{60} + 64 = 50$

Step 1.7

Multiply $60$ by $- 16$ .

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 16}{60} + 64 = 50$

Step 1.8

To write $64$ as a fraction with a common denominator, multiply by $\frac{60}{60}$ .

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 16}{60} + 64 \cdot \frac{60}{60} = 50$

Step 1.9

Simplify terms.

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Step 1.9.1

Combine $64$ and $\frac{60}{60}$ .

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 16}{60} + \frac{64 \cdot 60}{60} = 50$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 16 + 64 \cdot 60}{60} = 50$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply $64$ by $60$ .

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 16 + 3840}{60} = 50$

Step 1.10.2

Add $16$ and $3840$ .

$\frac{- 960 x + 60 x^{2} + y^{2} + 8 y + 3856}{60} = 50$

Step 1.11

Simplify with factoring out.

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Step 1.11.1

Factor $- 1$ out of $- 960 x$ .

$\frac{- (960 x) + 60 x^{2} + y^{2} + 8 y + 3856}{60} = 50$

Step 1.11.2

Factor $- 1$ out of $60 x^{2}$ .

$\frac{- (960 x) - (- 60 x^{2}) + y^{2} + 8 y + 3856}{60} = 50$

Step 1.11.3

Factor $- 1$ out of $- (960 x) - (- 60 x^{2})$ .

$\frac{- (960 x - 60 x^{2}) + y^{2} + 8 y + 3856}{60} = 50$

Step 1.11.4

Factor $- 1$ out of $y^{2}$ .

$\frac{- (960 x - 60 x^{2}) - 1 (- y^{2}) + 8 y + 3856}{60} = 50$

Step 1.11.5

Factor $- 1$ out of $- (960 x - 60 x^{2}) - 1 (- y^{2})$ .

$\frac{- (960 x - 60 x^{2} - y^{2}) + 8 y + 3856}{60} = 50$

Step 1.11.6

Factor $- 1$ out of $8 y$ .

$\frac{- (960 x - 60 x^{2} - y^{2}) - (- 8 y) + 3856}{60} = 50$

Step 1.11.7

Factor $- 1$ out of $- (960 x - 60 x^{2} - y^{2}) - (- 8 y)$ .

$\frac{- (960 x - 60 x^{2} - y^{2} - 8 y) + 3856}{60} = 50$

Step 1.11.8

Rewrite $3856$ as $- 1 (- 3856)$ .

$\frac{- (960 x - 60 x^{2} - y^{2} - 8 y) - 1 \cdot - 3856}{60} = 50$

Step 1.11.9

Factor $- 1$ out of $- (960 x - 60 x^{2} - y^{2} - 8 y) - 1 (- 3856)$ .

$\frac{- (960 x - 60 x^{2} - y^{2} - 8 y - 3856)}{60} = 50$

Step 1.11.10

Simplify the expression.

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Step 1.11.10.1

Rewrite $- (960 x - 60 x^{2} - y^{2} - 8 y - 3856)$ as $- 1 (960 x - 60 x^{2} - y^{2} - 8 y - 3856)$ .

$\frac{- 1 (960 x - 60 x^{2} - y^{2} - 8 y - 3856)}{60} = 50$

Step 1.11.10.2

Move the negative in front of the fraction.

$- \frac{960 x - 60 x^{2} - y^{2} - 8 y - 3856}{60} = 50$

Step 2

Multiply both sides by $60$ .

$- \frac{960 x - 60 x^{2} - y^{2} - 8 y - 3856}{60} \cdot 60 = 50 \cdot 60$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- \frac{960 x - 60 x^{2} - y^{2} - 8 y - 3856}{60} \cdot 60$ .

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Step 3.1.1.1

Cancel the common factor of $60$ .

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Step 3.1.1.1.1

Move the leading negative in $- \frac{960 x - 60 x^{2} - y^{2} - 8 y - 3856}{60}$ into the numerator.

$\frac{- (960 x - 60 x^{2} - y^{2} - 8 y - 3856)}{60} \cdot 60 = 50 \cdot 60$

Step 3.1.1.1.2

Cancel the common factor.

$\frac{- (960 x - 60 x^{2} - y^{2} - 8 y - 3856)}{60} \cdot 60 = 50 \cdot 60$

Step 3.1.1.1.3

Rewrite the expression.

$- (960 x - 60 x^{2} - y^{2} - 8 y - 3856) = 50 \cdot 60$

Step 3.1.1.2

Apply the distributive property.

$- (960 x) - (- 60 x^{2}) - - y^{2} - (- 8 y) - - 3856 = 50 \cdot 60$

Step 3.1.1.3

Simplify.

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Step 3.1.1.3.1

Multiply $960$ by $- 1$ .

$- 960 x - (- 60 x^{2}) - - y^{2} - (- 8 y) - - 3856 = 50 \cdot 60$

Step 3.1.1.3.2

Multiply $- 60$ by $- 1$ .

$- 960 x + 60 x^{2} - - y^{2} - (- 8 y) - - 3856 = 50 \cdot 60$

Step 3.1.1.3.3

Multiply $- - y^{2}$ .

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Step 3.1.1.3.3.1

Multiply $- 1$ by $- 1$ .

$- 960 x + 60 x^{2} + 1 y^{2} - (- 8 y) - - 3856 = 50 \cdot 60$

Step 3.1.1.3.3.2

Multiply $y^{2}$ by $1$ .

$- 960 x + 60 x^{2} + y^{2} - (- 8 y) - - 3856 = 50 \cdot 60$

Step 3.1.1.3.4

Multiply $- 8$ by $- 1$ .

$- 960 x + 60 x^{2} + y^{2} + 8 y - - 3856 = 50 \cdot 60$

Step 3.1.1.3.5

Multiply $- 1$ by $- 3856$ .

$- 960 x + 60 x^{2} + y^{2} + 8 y + 3856 = 50 \cdot 60$

Step 3.1.1.4

Move $- 960 x$ .

$60 x^{2} + y^{2} - 960 x + 8 y + 3856 = 50 \cdot 60$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $50$ by $60$ .

$60 x^{2} + y^{2} - 960 x + 8 y + 3856 = 3000$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract $3000$ from both sides of the equation.

$60 x^{2} + y^{2} - 960 x + 8 y + 3856 - 3000 = 0$

Step 4.2

Subtract $3000$ from $3856$ .

$60 x^{2} + y^{2} - 960 x + 8 y + 856 = 0$